Expression of type Conditional

from the theory of proveit.physics.quantum.QPE

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, k, m
from proveit.linear_algebra import ScalarMult
from proveit.logic import Equals, InSet
from proveit.numbers import Exp, Mult, e, i, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _phase, _t

# build up the expression from sub-expressions
sub_expr1 = NumBra(m, _t)
sub_expr2 = InverseFourierTransform(_t)
sub_expr3 = NumKet(k, _t)
sub_expr4 = Exp(e, Mult(two, pi, i, _phase, k))
expr = Conditional(Equals(Qmult(sub_expr1, sub_expr2, ScalarMult(sub_expr4, sub_expr3)), Mult(sub_expr4, Qmult(sub_expr1, sub_expr2, sub_expr3))), InSet(k, _m_domain))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{\left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \lvert k \rangle_{t}\right)\right) \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operands: 4
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7, 8
5	Literal
6	ExprTuple	52, 9
7	Operation	operator: 20 operands: 10
8	Operation	operator: 46 operands: 11
9	Operation	operator: 12 operands: 13
10	ExprTuple	25, 26, 14
11	ExprTuple	24, 15
12	Literal
13	ExprTuple	16, 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 20 operands: 21
16	Literal
17	Operation	operator: 22 operands: 23
18	Literal
19	ExprTuple	24, 27
20	Literal
21	ExprTuple	25, 26, 27
22	Literal
23	ExprTuple	28, 29
24	Operation	operator: 37 operands: 30
25	Operation	operator: 31 operands: 32
26	Operation	operator: 33 operand: 44
27	Operation	operator: 35 operands: 36
28	Operation	operator: 37 operands: 38
29	Operation	operator: 39 operand: 45
30	ExprTuple	41, 42
31	Literal
32	ExprTuple	43, 44
33	Literal
34	ExprTuple	44
35	Literal
36	ExprTuple	52, 44
37	Literal
38	ExprTuple	48, 44
39	Literal
40	ExprTuple	45
41	Literal
42	Operation	operator: 46 operands: 47
43	Variable
44	Literal
45	Literal
46	Literal
47	ExprTuple	48, 49, 50, 51, 52
48	Literal
49	Literal
50	Literal
51	Literal
52	Variable